32-6 Diamagnetism#

Prompts

For a diamagnetic sample in an external magnetic field, what is the direction of the induced dipole moment relative to the field? Why?
In a nonuniform magnetic field, describe the force on a diamagnetic sample and the resulting motion. How does this differ from paramagnetic materials?
Compare diamagnetism, paramagnetism, and ferromagnetism. For each, describe the response to an external field and give an example.

Lecture Notes#

Overview#

All materials exhibit weak diamagnetism, but it is masked if paramagnetism or ferromagnetism is present
Diamagnetic materials: exhibit only diamagnetism (e.g., bismuth, copper, water)
Induced dipoles: Placed in \(\vec{B}_{\text{ext}}\), atoms develop dipole moments opposite to the field
No permanent moments: Dipoles disappear when \(\vec{B}_{\text{ext}}\) is removed

Physical intuition

Changing \(\vec{B}\) induces currents in electron orbits (Faraday’s law). The induced field opposes the change (Lenz’s law), so the induced dipole moment is opposite to \(\vec{B}_{\text{ext}}\).

Torque and force#

A magnetic dipole \(\vec{\mu}\) in a field \(\vec{B}\) experiences torque \(\vec{\tau} = \vec{\mu} \times \vec{B}\) (tends to align \(\vec{\mu}\) with or oppose \(\vec{B}\)) and orientation energy \(U = -\vec{\mu} \cdot \vec{B}\). In a nonuniform field, apart from the torque, a net force also arises from the gradient; its direction depends on whether \(\vec{\mu}\) aligns with or opposes \(\vec{B}\).

Material	\(\vec{\mu}\) vs \(\vec{B}\)	Uniform field	Nonuniform field
Diamagnetic	\(\vec{\mu}\propto -\vec{B}\)	Torque aligns \(\vec{\mu}\) oppose \(\vec{B}\)	Repelled from stronger field
Paramagnetic, Ferromagnetic	\(\vec{\mu}\propto \vec{B}\)	Torque aligns \(\vec{\mu}\) with \(\vec{B}\)	Attracted to stronger field

How a magnet senses the field gradient

1. Loop model: Treat the induced (or permanent) current in an electron orbit as a current loop. In a diverging (nonuniform) field, the magnetic force \(\vec{F} = i\,\vec{L} \times \vec{B}\) on opposite sides of the loop does not cancel. The direction of the net force depends on the dipole orientation:

Diamagnetic: When \(\vec{\mu}\) is opposite to \(\vec{B}\), the net force points against the stronger field.
Paramagnetic, Ferromagnetic: When \(\vec{\mu}\) aligns with \(\vec{B}\), the net force points toward the stronger field.

2. Energy argument: The orientation energy is \(U = -\vec{\mu} \cdot \vec{B}\). Along the field direction, \(U = -\mu B\) (taking \(\mu\) and \(B\) as signed scalars; for diamagnetic, \(\mu < 0\) because \(\vec{\mu}\) opposes \(\vec{B}\)). The force is \(F = -\frac{dU}{ds} = \mu\,\frac{dB}{ds}\). Assuming \(dB/ds > 0\) (field increases toward stronger region):

Diamagnetic: When \(\mu < 0\), \(F < 0\) — the force points against the stronger field.
Paramagnetic, Ferromagnetic: When \(\mu > 0\), \(F > 0\) — the force points toward the stronger field.

Summary#

Diamagnetism: Induced dipoles opposite to \(\vec{B}_{\text{ext}}\); weak; present in all materials
Diamagnetic materials: Repelled from stronger field
Torque and force (see table above): Same torque and energy laws for dia/para/ferro; force in nonuniform field differs by dipole orientation
Diamagnetism often masked by paramagnetism or ferromagnetism when those are present

Discussions#

Multipole definitions#

A monopole is a single isolated source or sink of field lines.

Electric: Point charge \(q\) (positive or negative).
Magnetic: Hypothetical magnetic charge (magnetic monopole). None observed.

A dipole is two equal-and-opposite monopoles separated by a small distance.

Electric dipole: Charges \(+q\) and \(-q\) separated by \(\vec{d}\); dipole moment \(\vec{p} = q\vec{d}\) (from \(-q\) to \(+q\)).
Magnetic dipole: North and south poles, or a current loop; dipole moment \(\vec{\mu}\) (direction S→N, or right-hand rule for loop).

A quadrupole is two dipoles with opposite moments (or four charges in symmetric arrangement). Higher multipoles (octupole, etc.) add faster-decaying terms in the multipole expansion.

Electric vs magnetic dipole analogy#

Property	Electric dipole	Magnetic dipole
Moment	\(\vec{p} = q\vec{d}\)	\(\vec{\mu}\) (current loop: \(\vec{\mu} = I\vec{A}\))
Potential energy	\(U = -\vec{p} \cdot \vec{E}\)	\(U = -\vec{\mu} \cdot \vec{B}\)
Force in uniform field	Zero	Zero
Force in nonuniform field	\(\vec{F} = (\vec{p} \cdot \nabla)\vec{E}\)	\(\vec{F} = (\vec{\mu} \cdot \nabla)\vec{B}\)
Produced field far away	\(E \propto 1/r^3\)	\(B \propto 1/r^3\)

Both share the same energy form \(U = -\vec{m} \cdot \vec{F}_{\text{field}}\) (with \(\vec{m}\) = moment, \(\vec{F}_{\text{field}}\) = \(\vec{E}\) or \(\vec{B}\)). The torque tends to rotate the dipole toward the lower-energy (parallel) orientation.